Dynamical behaviors of a stochastic SIRI epidemic model

Abstract

In this paper, we initially propose a stochastic SIRI epidemic model. The existence, uniqueness, and boundedness of the positive solution of the stochastic system are derived. By using some novel Lyapunov functions and stochastic analysis tools, we mainly investigate asymptotical behaviors of the SIRI model under stochastic perturbations according to the basic reproduction number . That is, when (or ) and some conditions on white noise are satisfied, the solution of stochastic model will oscillate around the disease-free equilibrium (or endemic equilibrium) of the deterministic model and the estimate for the oscillation amplitude is obtained. Furthermore, the existence of a stationary distribution and the ergodicity of solutions are studied for the stochastic system. Finally, some numerical examples are provided to illustrate the effectiveness of the obtained results.

Keywords:

• Stochastic SIRI epidemic model
• asymptotic behavior
• stationary distribution
• ergodicity
• stability

AMS Subject Classifications:

• 34D20
• 34K20
• 92B50

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported partially by National Natural Science Foundation of China under [grant number 11471061]; the Natural Science Foundation of Chongqing under [grant number CQ CSTC 2014JCYJA40004]; the Program of Chongqing Innovation Team Project in University under [grant number KJTD201308]; the Postgraduate Scientific Innovation Project of Chongqing Normal University under [grant number YKC15015].